Abstract
In this game (Aruka 2001), selfishness may not be determined even if an agent selfishly adopts the strategy of defection. Individual selfishness can only be realized if the other agent cooperates, therefore gain from defection can never be assured by defection alone. The sanction by defection as a reaction of the rival agent cannot necessarily reduce the selfishness of the rival. In this game, explicit direct reciprocity cannot be guaranteed. Now we introduce different spillovers or payoff matrices, so that each agent may then be faced with a different payoff matrix. A ball in the urn is interpreted as the number of cooperators, and the urn as a payoff matrix. We apply Ewens’ sampling formula to our urn process in this game theoretic environment.
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- 1.
See Fujiwara (2008).
- 2.
This figure is almost equivalent to Figure 3 in Aruka (2007).
- 3.
This interpretation is supplemented by citing Bowles (2004), in terms of behavioral dynamics: “Thus individuals are the bearers of behavioral rules. Analytical attention is focused on the success or failure of these behavioral rules themselves as they either diffuse and become pervasive in a population or fail to do so and are confined to minor ecological niches or are eliminated. The dramatis personae of the social dynamic thus are not individuals but behavioral rules: how they fare is the key; what individuals do is important for how this contributes to the success or failure of behavioral rules”.
- 4.
Hildenbrand (1994) called it “macroscopic microeconomic” linkage.
- 5.
This was pointed out by an anonymous referee to whom we are indebted for this description.
- 6.
The risk may be the risk of realized expectation, whether the expectation is cooperation or defection, when agents wander around.
- 7.
As Bowles (2004, p. 62) argued, this technique is not superficial: “What is called matching noise is another way that chance affects evolutionary dynamics. When small numbers of individuals in a heterogeneous population are randomly paired to interact, the realized distribution of types with whom one is paired over a given period may diverge significantly from the expected distribution. The difference between the realized distribution and the expected distribution reflects matching noise and may have substantial effects. … How, then, are evolutionary models different? First, mutations, behavioral innovations, and matching noise are distinct because these sources of stochastic events are endogenous to evolutionary models”.
- 8.
D denotes “Defection” while C denotes “Cooperation”.
- 9.
This statement needs an assumption like Assumption 3, given in Section 4.
- 10.
We can add to a more general proposition: in non-balanced triangular urn models as depending on the values of parameters, no-self-averaging emerges (Aoki and Yoshikawa 2007, p. 14).
- 11.
Aoki and Yoshikawa (2007, p. 6) argued the economic meaning of non-self-averaging as follows: the notion of non-self-averaging is important because non-self-averaging models are sample dependent, and some degree of impreciseness or dispersion remains about the time trajectories even when the number of economic agents goes to infinity. This implies that a focus on the mean path behavior of macroeconomic variables is not justified. It, in turn, means that sophisticated optimization exercises that provide us with information on the means have little value.
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Aruka, Y., Akiyama, E. (2011). Non-Self-Averaging of a Two-Person Game with Only Positive Spillover: A New Formulation of Avatamsaka’s Dilemma. In: Aruka, Y. (eds) Complexities of Production and Interacting Human Behaviour. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2618-0_12
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